Technically a nation is just a large collective of people that agree that their they are in a group relationship with each other due to their common, language, heritage, culture, customs, tradition or whatnot.
So they exist as long as they affirm this group relation and they would seize to exist if the individuals constituting it no longer do that or no longer exist. That alone usually wouldn't require a "right to exist" in the first place as it doesn't interfere with other people outside of that group.
So what you probably refer to is likely not a "nation", but a "nation STATE". That is a nation that claims for itself the right to self-determination, usually including a plot of land of their own in which they can exercise political autonomy.
So I'm not so sure about that one:
Throughout human history, at any point in time, the Earth has been a mixed bag of nations that were either peaceful towards its neighbors and nations that were aggressive towards their neighbors.
Tribes have existed for a long time, but it's not self-evident that a tribe necessarily has it's own state/country (roaming people) or political autonomy (national minorities within a state) and states/empires have existed for quite some time but it's not self-evident that they identified as one nation or as a group at all and not just as property of the king/aristocracy. But the idea of nationalism and nation states is as far as I know a pretty recent one.
The other problem is that "right to ..." is a meaningless term without a context, because rights are agreements between individuals within an explicit or implicit group. So in order for rights to make sense you need other entities that could honor or violate these rights and they must be able to understand and agree with these rights otherwise it's meaningless waste of paper and just a survival of the fittest (not the strongest but whatever works best at staying alive and autonomous in the presence of other entities that either don't care about that or in the worst case are actively hostile to that idea).
And if you enter the stage of supernational or international law things get really complicated because there is no superior authority being able to dictate and enforce laws and neither do these different factions consider themselves one group that mutually wants to stick together, so it's a complicated mess of agreements that largely only hold value as long as they are mutually agreed upon because if one side thinks they violate their interests they could technically just leave the contract and who's going to stop them?
So the best you can do against such destructive behavior by individual states is to form a union of states that codifies mutually agreed upon laws and expels rule breakers or threatens collective actions against them in case of rule breaking. The problem is that this is a double edged sword, on the one hand it's great for upholding mutually agreed upon rules, on the other hand states like to leverage the power of that group while simultaneously fear they could end up on the receiving end of that power. So often enough institutions like the United Nations are rather tame with regards to their ability to punish the behavior of states. Also the the power stems from their size so they like to incorporate most if not all nations and as a consequence don't can't afford to expel members so that their acceptance for behavior is huge and their power to restrict is often limited. Not to mention that larger states still play by different rules than smaller states. Like the permanent members of the security council can fight illegal wars and block resolutions condemning that and if you can form blocks of states and alliances that are effective enough on their own to not require the rest of the states you could ignore their legislation altogether.
So international law is somewhere between anarchism (leaderless coexistence/cooperation) and chaos and anomie (survival of the fittest/might makes right etc) .
And with regards to who is allowed to join these clubs of states. Well... That's actually not at all clear to begin with. It's not even clear what IS a state. Like there are definitions like you'd need a land, a population and a government to be a state, while others argue that they are states and if they they you are one two then you are one, there are countries that recognize other countries as such while others deny them the respective status, there are de facto states that de jure aren't and vice versa. And technically you can just assert that status and if no one challenges it you'll likely become a state by tradition at one point.
Like what if a nation within a state decides to secede from that state and form it's own nation state? Is that a self-determination of a new nation or is that an act of criminality that the existing state has the right to determine by itself and to surpress if they see fit? Both could argue with "self-determination".
And technically a warring nation could be ousted, condemned and punished but practically that depends on it's size and how many will join that. Also while on paper the "just war theory" is a nice thing (arguing that only self-defense and the like are just reasons to go to war). Just look at the Nazis and you'll find that a fanatically racist country could just make up a boogeyman of which it is so damn threatened that they formulate a right to proactive "self-defense" in other words a war of aggression with dimensions of a genocide. It's a very dubious claim to be made, but it nonetheless fulfills the technical limits of "self-defense" even if the threat is just a subjective feeling. If there is no objective institution to judge it comes down to the winners and fingers crossed it's not the fanatic morons.